arXiv subjects

Tiziano Penati

Publications and source records attributed to Tiziano Penati.

11 recordsLinked to original sources

arXiv2024

Coreo-Graph: un progetto di teoria dei grafi e danza

In this contribution we revisit a learning laboratory of graph theory, based on the well-known Königsberg's bridges problem due to Euler, already proposed and described in Gaio-Capone-Branchetti (2020): we extend some of its conceptual aspects and laboratory activities, in particular by inserting a dance-inspired activity in order to explore the properties of Eulerian paths. In this new guise, the project was proposed and tested in the period 2022-2024 as an activity to strengthen the curricular skills of problem-solving. At the same time, it was an opportunity to think over the role of embodied activities in facilitating understanding and assimilation of abstract concepts of mathematics.

math.HO↗

arXiv2024

Darboux's Theorem, Lie series and the standardization of the Salerno and Ablowitz-Ladik models

In the framework of nonlinear Hamiltonian lattices, we revisit the proof of Moser-Darboux's Theorem, in order to present a general scheme for its constructive applicability to Hamiltonian models with non-standard symplectic structures. We take as a guiding example the Salerno and Ablowitz-Ladik (AL) models: we justify the form of a well-known change of coordinates which is adapted to the Gauge symmetry, by showing that it comes out in a natural way within the general strategy outlined in the proof. Moreover, the full or truncated Lie-series technique in the extended phase-space is used to transform the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS Hamiltonian turns out to be a normal form of the Salerno and AL models; as a byproduct we also get estimates of the dynamics of these models by means of dNLS one. We also stress that, once it is cast into the perturbative approach, the method allows to deal with the cases where the explicit trasformation is not known, or even worse it is not writable in terms of elementary functions.

math-ph↗

arXiv2021

Continuation of spatially localized periodic solutions in discrete NLS lattices via normal forms

We consider the problem of the continuation with respect to a small parameter $ε$ of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where $ε$ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localised periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.

math.DS↗

arXiv2020

On the continuation of degenerate periodic orbits via normal form: lower dimensional resonant tori

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.

math.DS↗

arXiv2019

Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit

In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$ in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength $\eps$. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order $\mathcal{O}(\exp(\eps^{-1}))$, is also obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.

math.DS↗

arXiv2017

On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori

We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.

math.DS↗

arXiv2015

Long time stability of small amplitude Breathers in a mixed FPU-KG model

In the limit of small couplings in the nearest neighbor interaction, and small total energy, we apply the resonant normal form result of a previous paper of ours to a finite but arbitrarily large mixed Fermi-Pasta-Ulam Klein-Gordon chain, i.e. with both linear and nonlinear terms in both the on-site and interaction potential, with periodic boundary conditions. An existence and orbital stability result for Breathers of such a normal form, which turns out to be a generalized discrete Nonlinear Schrödinger model with exponentially decaying all neighbor interactions, is first proved. Exploiting such a result as an intermediate step, a long time stability theorem for the true Breathers of the KG and FPU-KG models, in the anti-continuous limit, is proven.

math.DS↗

arXiv2014

An extensive resonant normal form for an arbitrary large Klein-Gordon model

We consider a finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions. In the limit of small couplings in the nearest neighbor interaction, and small (total or specific) energy, a high order resonant normal form is constructed with estimates uniform in the number of degrees of freedom. In particular, the first order normal form is a generalized discrete nonlinear Schroedinger model, characterized by all-to-all sites coupling with exponentially decaying strength.

math.DS↗

arXiv2014

An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit

We construct an extensive adiabatic invariant for a Klein-Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant $a$, the evolution of the adiabatic invariant is controlled up to times scaling as $β^{1/\sqrt{a}}$ for any large enough value of the inverse temperature $β$. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.

math.DS↗

arXiv2014

Normal Form and Energy Conservation of High Frequency Subsystems Without Nonresonance Conditions

We consider a system in which some high frequency harmonic oscillators are coupled with a slow system. We prove that up to very long times the energy of the high frequency system changes only by a small amount. The result we obtain is completely independent of the resonance relations among the frequencies of the fast system. More in detail, denote by $ε^{-1}$ the smallest high frequency. In the first part of the paper we apply the main result of [BG93] to prove almost conservation of the energy of the high frequency system over times exponentially long with $ {ε^{-1/n}} $ ($n$ being the number of fast oscillators). In the second part of the paper we give e new self-contained proof of a similar result which however is valid only over times of order $ε^{-N}$ with an arbitrary $N$. Such a second result is very similar to the main result of the paper [GHL13], which actually was the paper which stimulated our work.

math.DS↗

arXiv2006

Tail resonances of FPU q-breathers and their impact on the pathway to equipartition

Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincare-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets.

nlin.PS↗